Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,436
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Question about the Kähler structure on generic coadjoint orbits
Let $G$ be a compact connected Lie group. We denote by $\mathfrak{g}$ the Lie algebra of $G$ and by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $\mathcal{O}_r: = G\cdot r$ be a generic ...
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Manifold whose symplectic structure of the cotangent bundle is intrinsically different from any symplectic structure arising from $\mathbb{C}^n$
Inspired by this question Symplectic structure of $TS^{n-1}$ we ask:
What is an example of a manifold $M$ whose cotangent bundle $T^*M$ is an Stein manifold but the canonical symplectic ...
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Size of Hilbert space in geometric quantization from index theorem
In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem.
To be precise, the polarization ...
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Relation between symplectic (co)homology and Hochschild (co)homology and deformations
A very fluffy question in which I'm ignorant of homology/cohomology, grading etc:
The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the ...
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Iterating exact triangles (particularly in Floer homology)
There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
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Realizing closed manifolds as Legendrian submanifolds of the standard contact vector space
I started learning some basic contact geometry, in particular its flexible side, and I got stuck with the following naive question. Given a closed manifold of dimension $n$, we can always embed it ...
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The norm-squared of a moment map behaves like a Morse-Bott function
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Let $<.,>$ denote a $G$-invariant inner product on $\mathfrak{g}$.
Let $(M,\omega)$ be a symplectic compact manifold endowed with ...
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Cotangent bundles to Riemannian manifolds and submanifolds
Let $N$ be a Riemannian manifold, and $\widetilde{N}\subset N$ a closed submanifold. If we look at the total spaces of the tangent bundles, we get that $T\widetilde{N}$ is a submanifold of $TN$.
If ...
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Holomorphic/Symplectic embedding of Riemann surfaces
Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$?
What about ...
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When does a holomorphic symplectic manifold compactify to a Poisson manifold?
Let $X$ be a complex manifold endowed with a holomorphic closed 2-form $\omega$ whose associated map $\omega : TX \to T^*X$ is invertible. Can we always embed $X$ as an open subset of a compact ...
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From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures
Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.
Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson ...
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Choice of a family of almost complex structures when defining Floer Homology
Consider a $1$-periodic Hamiltonian $H:S^{1}\times M\rightarrow \mathbb{R}$ defined on a compact symplectic manifold $M$. Let's suppose $M$ is nice enough so that we can develop Floer theory on it. ...
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Generic choice of non-degenerate Hamiltonians $H$ in Floer theory
When developing floer theory for an Hamiltonian $H:M\times S^{1}\rightarrow \mathbb{R}$ we will want $H$ to satisfy a non-degenerancy condition, that is, for every $x\in \mathcal{P}(H)$, a periodic ...
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Are all exact Lagrangian spheres, vanishing cycles?
Let $\pi: E \to D$ be an exact Lefschetz fibration with corners (fibers with boundary)over the disk. Fix a point $\theta \in \partial D$ and consider the fiber $F_\theta = \pi^{-1}(\theta)$ over that ...
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Definition of Floer complex in Floer's "Morse theory for Lagrangian intersections"
I am moving the first steps into Lagrangian Floer theory and I am trying to understand the construction, as in the original paper, for the field $\mathbb{Z}_2$ (no orientations) and $\pi_2(P,L) = 0$.
...
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What is the interpretation of Jacobi Identity on sympletic manifold?
Context (pg-321):
We have a manifold with an anti symmetric metric tensor/sympletic form $S$ with components in a basis $S_{ab}$ satisfying the property that
$$dS=0$$
Where $d$ is the exterior ...
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symplectic gromov witten invariants of weighted projective space
Does anyone know if the symplectic (genus zero) GW invariants have been computed for weighted projective spaces? It seems there are already algebraic computations https://arxiv.org/abs/math/0608481
Is ...
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On exotic symplectic structures of smooth closed 4-manifolds
What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
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Existence of non-trivial "line-symplectic" manifolds
One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\...
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Linearization of the Floer equation
In Floer Homology we want to prove that the Moduli spaces $\mathcal{M}(x^{-},x^{+})$ are finite dimensional manifolds. This is done by expressing them as the zero set of a Fredholm map. First one ...
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Gromov width of weighted projective planes
I am interested in knowing the Gromov width of (the complement of the three orbifold points of) weighted projective planes $\Bbb{CP}(a,b,c)$. Let me emphasize that I am mainly interested in upper ...
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Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?
This is a crosspost from this MSE question from a year ago.
Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form?
If $\omega$ ...
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Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-holomorphic curves necessarily symplectomorphism?
Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover ...
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Gluing of Morse-type trajectories in "Floer Homology of Cotangent bundles"
In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
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Gluing of hybrid trajectories in Floer homology
In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
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Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$
I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$.
Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...
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Symplectic cohomology of $T^* \mathbb{CP}^2$
I'm looking for an explanation for why the symplectic cohomology $SH^*(T^* \mathbb{CP}^2,\mathbb{Z})$ is 2-torsion (I heard this in passing; perhaps it's not even true!). By a clever argument that I ...
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Results proved using the geometry of moduli spaces of pseudo-holomorphic curves?
It is well-known that the Gromov-Witten invariants and their Floer-theoretic counterpart of symplectic manifolds have rich algebraic structures. However, sometimes it's quite useful even by ...
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Chern number for curves in a toric variety
Suppose $X$ is a non-singular toric variety that is the closure of the complex torus $X^\circ \simeq T_{\mathbb C}$. Let $Y_1,\dots, Y_N$ be the closures of codimension one $T_{\mathbb C}$-orbits, so ...
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Motivation behind the usual setting of the (weak) Arnold conjecture for fixed points of an hamiltonian diffeomorphism
I'm trying to find out the motivations that led V. Arnold to formulate his famous conjecture (I guess theorem by now) in the following form:
Let $(M,\omega)$ be a closed symplectic manifold (add ...
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Multisymplectic connections and topological invariants
I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry.
From symplectic connections, it is well ...
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Charts for the Banach manifold of smooth almost complex structures $\mathcal{J}^{l}$
Consider the closure in the $C^l$-topology of the space of smooth almost complex structrues of a symplectic manifold $(M,\omega)$. We will denote this space by $\mathcal{J}^l$.
It's a very used fact ...
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$\omega$ is a symplectic form then $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$
Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$. Let $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ be the map given by $L^k_{\omega}(\alpha)=\alpha\wedge\omega^k$. Then is it true ...
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plumbing description of resolution of ADE singularities
Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $...
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Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic
Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G_\beta$ the stabilizer subgroup of $\beta$ by ...
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Sectional curvature and injectivity radius of natural metric in cotangent bundles
In the following paper by Cielibak, Ginzburg and Kerman (arXiv link, Comm. Math. Helv. 2004 DOI link) they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is ...
user174565
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Proving that a twisted cotangent bundle is geometrically bounded
In Cieliebak, Ginzburg and Kerman's paper Symplectic homology and periodic orbits near symplectic submanifolds, the authors claim and give a proof that a twisted cotangent bundle will be geometrically ...
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Fujiki class $\mathcal C$ with a symplectic structure
Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ ...
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Influence of symplectic invariants of the complement on being superheavy
Let $(M,\omega)$ be a symplectic manifold.
I'm trying to show that a compact subset $K\subset M$ is $1$-superheavy [1, Definition 1.3] where $1=PD([M])$ is the unit in $QH^0(M)$.
My question is: How ...
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If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?
Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
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Cup product and PSS map
Let $(M,\omega)$ be a symplectic manifold and let $H$ a Hamiltonian function. If $M$ is not closed we consider $H$ to be linear at infinity to ensure that $HF^*(H)$ is well-defined (I'm particularly ...
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From Delzant polytope to lattice polytope
By definition, an $n$-dimensional Delzant polytope $P$ is not necessarily a lattice polytope. But
is there a natural way (or operations) to turn $P$ into a lattice polytope using the fact that the ...
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Continuation principle for solutions of Floer's equation in $\mathbb{R}\times [0,1]$ and transversality
Consider $(M,\omega)$ a symplectic manifold and $J$ a compatible almost complex structure. For me it's well known that if we consider 2 solutions $u,v:\mathbb{R}\times S^1\rightarrow M$ of Floer's ...
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Symplectic orbits in projective Hilbert spaces are simply connected
Let $G$ be a connected Lie group and let $(\pi, \mathcal{H})$ be an irreducible unitary representation of $G$ on an infinite-dimensional Hilbert space $\mathcal{H}$. Denote by $\mathcal{H}^{\infty}$ ...
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Writting the Floer map in local coordinates using the exponential chart
Following Salamon's Notes in Floer Homology , consider the Floer equation
$$\mathcal{F}(u):=\partial_su+J_t(\partial_tu+\nabla H_t(u))=0$$
Then we can write in local coordinates
$$\mathcal \Phi_u^{-1}(...
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Proof of the Hamiltonian slice theorem
Theorem(Guillemin Sternberg Marle) Let $(M, \omega, \mu) $ be a symplectic manifold together with a Hamiltonian group action. Let $p$ be a point in $M$ such that $O_p $ is contained in the zero level ...
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Compatible almost complex structures such that the associated riemannian metric has positive injectivity radius
Let $M$ be a compact manifold, consider $\omega$ the canonical symplectic form in $T^*M$ and $\hat J$ the canonical almost complex structure coming from the Sasaki metric.
Let $\mathcal{J}$ be the set ...
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Which symplectic manifolds are coadjoint orbits of finite-dimensional $G$ on $\mathfrak{g}^*$ with the Kostant–Kirillov–Souriau Poisson structure?
Given some finite-dimensional symplectic manifold $(M,\omega)$, can we answer whether or not it arises as the coadjoint orbit of some finite-dimensional $G$ acting on the dual $\mathfrak{g}^*$ to its ...
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Dismissing pseudoholomorphic curves in embedded contact homology
In the papers
The periodic Floer homology of a Dehn twist,
Rounding corners of polygons and the embedded contact homology of $T^3$,
and Combinatorial embedded contact homology for toric contact ...
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Associativity of orientations of determinant bundles in Floer homology
I have been reading the paper "Coherent orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer, trying to understand how we can orient the moduli spaces that ...
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